Rank three representations of Painlevé systems: II. de Rham structure, Fourier--Laplace transformation

TL;DR

This work develops a rigorous bridge between linear D-module representations and nonlinear Painlevé dynamics by employing formal microlocalization to analyze the Fourier--Laplace transform between rank-$3$ JKT representations and rank-$2$ irregular connections. The authors show that, under suitable parameter choices, the Fourier--Laplace transform induces a biregular morphism between the de Rham moduli spaces $\mathcal{M}_{dR}^{\text{JKT}*}$ and the corresponding Painlevé moduli, with explicit local-form analyses confirming rank reductions and the appearance of Painlevé phase spaces (I–IV) as FL images. The approach leverages six local models and the formal stationary phase principle to determine irregular types and singularity structures, providing a robust algebraic route to non-abelian Hodge-type correspondences for Painlevé systems. The results advance the understanding of how geometric Fourier transforms encode isomonodromic data and wild character varieties, with potential generalization to broader non-abelian Hodge settings and higher-rank correspondences.

Abstract

We use formal microlocalization to describe the Fourier--Laplace transformation between rank 3 and rank 2 D-module representations of Painleve systems. We conclude the existence of biregular morphism between the corresponding de Rham complex structures.

Theorems & Definitions (19)

• Theorem 1.1: Proposition \ref{['prop:dRisomorphism']}
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Proposition 3.1
• Proposition 3.2
• Lemma 3.3
• Lemma 3.4
• proof